Step |
Hyp |
Ref |
Expression |
1 |
|
plusffval.1 |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
plusffval.2 |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
plusffval.3 |
⊢ ⨣ = ( +𝑓 ‘ 𝐺 ) |
4 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = ( Base ‘ 𝐺 ) ) |
5 |
4 1
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = 𝐵 ) |
6 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( +g ‘ 𝑔 ) = ( +g ‘ 𝐺 ) ) |
7 |
6 2
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( +g ‘ 𝑔 ) = + ) |
8 |
7
|
oveqd |
⊢ ( 𝑔 = 𝐺 → ( 𝑥 ( +g ‘ 𝑔 ) 𝑦 ) = ( 𝑥 + 𝑦 ) ) |
9 |
5 5 8
|
mpoeq123dv |
⊢ ( 𝑔 = 𝐺 → ( 𝑥 ∈ ( Base ‘ 𝑔 ) , 𝑦 ∈ ( Base ‘ 𝑔 ) ↦ ( 𝑥 ( +g ‘ 𝑔 ) 𝑦 ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + 𝑦 ) ) ) |
10 |
|
df-plusf |
⊢ +𝑓 = ( 𝑔 ∈ V ↦ ( 𝑥 ∈ ( Base ‘ 𝑔 ) , 𝑦 ∈ ( Base ‘ 𝑔 ) ↦ ( 𝑥 ( +g ‘ 𝑔 ) 𝑦 ) ) ) |
11 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
12 |
2
|
fvexi |
⊢ + ∈ V |
13 |
12
|
rnex |
⊢ ran + ∈ V |
14 |
|
p0ex |
⊢ { ∅ } ∈ V |
15 |
13 14
|
unex |
⊢ ( ran + ∪ { ∅ } ) ∈ V |
16 |
|
df-ov |
⊢ ( 𝑥 + 𝑦 ) = ( + ‘ 〈 𝑥 , 𝑦 〉 ) |
17 |
|
fvrn0 |
⊢ ( + ‘ 〈 𝑥 , 𝑦 〉 ) ∈ ( ran + ∪ { ∅ } ) |
18 |
16 17
|
eqeltri |
⊢ ( 𝑥 + 𝑦 ) ∈ ( ran + ∪ { ∅ } ) |
19 |
18
|
rgen2w |
⊢ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 + 𝑦 ) ∈ ( ran + ∪ { ∅ } ) |
20 |
11 11 15 19
|
mpoexw |
⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + 𝑦 ) ) ∈ V |
21 |
9 10 20
|
fvmpt |
⊢ ( 𝐺 ∈ V → ( +𝑓 ‘ 𝐺 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + 𝑦 ) ) ) |
22 |
|
fvprc |
⊢ ( ¬ 𝐺 ∈ V → ( +𝑓 ‘ 𝐺 ) = ∅ ) |
23 |
|
fvprc |
⊢ ( ¬ 𝐺 ∈ V → ( Base ‘ 𝐺 ) = ∅ ) |
24 |
1 23
|
eqtrid |
⊢ ( ¬ 𝐺 ∈ V → 𝐵 = ∅ ) |
25 |
24
|
olcd |
⊢ ( ¬ 𝐺 ∈ V → ( 𝐵 = ∅ ∨ 𝐵 = ∅ ) ) |
26 |
|
0mpo0 |
⊢ ( ( 𝐵 = ∅ ∨ 𝐵 = ∅ ) → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + 𝑦 ) ) = ∅ ) |
27 |
25 26
|
syl |
⊢ ( ¬ 𝐺 ∈ V → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + 𝑦 ) ) = ∅ ) |
28 |
22 27
|
eqtr4d |
⊢ ( ¬ 𝐺 ∈ V → ( +𝑓 ‘ 𝐺 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + 𝑦 ) ) ) |
29 |
21 28
|
pm2.61i |
⊢ ( +𝑓 ‘ 𝐺 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + 𝑦 ) ) |
30 |
3 29
|
eqtri |
⊢ ⨣ = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + 𝑦 ) ) |